Sets of all periodic points of a toral automorphism

In this article, subsets of T² that can arise as sets of all periodic points of a continuous 2-dimensional toral automorphism are characterized. Here the torus T² is viewed as [0,1)×[0,1) as a group under coordinatewise addition modulo 1.     

Construction of some wreath products as Galois groups of normal real extensions of the rationals

Let K be a real algebraic number field. Suppose that G occurs as a Galois group of a normal real extension field of K. Using elementary methods, we show that certain types of split extensions of an elementary abelian 2-group by G also occur as Galois groups of normal real extensions of K. Among other examples, we show that Sylow 2-subgroups of the symmetric and alternating groups of degree 2ⁿ, as well as the Weyl groups of type B_n and D_n, occur as Galois groups of real extensions of the rationals.     

Evaluation of the expected makespan of a set of non-resumable jobs on parallel machines with stochastic failures

In this paper we consider a production process at operative level on m$m$ identical parallel machines, which are subject to stochastic machine failures. To avoid long downtime of the machines, caused by unexpected failures, preventive maintenance activities are planned and conducted, but if a failure could not be averted a corrective maintenance has to be performed. Both maintenance activities are assumed to restore the machine to be “as good as new”. The maintenance activities, the number of jobs and their allocation to machines as well as their sequence have a large impact on the performance of the production process and the delivery dates.     

Nonexistence of Positive Supersolutions of Elliptic Equations via the Maximum Principle

We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of ? ⁿ . The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the p-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.     

The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation

We investigate a semilinear elliptic equation with a logistic nonlinearity and an indefinite nonlinear boundary condition, both depending on a parameter λ. Overall, we analyze the effect of the indefinite nonlinear boundary condition on the structure of the positive solutions set. Based on variational and bifurcation techniques, our main results establish the existence of three nontrivial non-negative solutions for some values of λ, as well as their asymptotic behavior. These results suggest that the positive solutions set contains an S-shaped component in some case, as well as a combination of a C-shaped and an S-shaped components in another case.     

Nonnull distributions of some statistics associated with testing for the equality of two covariance matrices

The nonnull distribution of some statistics, used for testing Σ₁ = Σ₂ are obtained as mixtures of incomplete beta functions as well as mixtures of incomplete gamma functions. The introduction of the convergence factors and certain recurrence relations are useful in the computation of the power of the tests as well as computation of exact percentage points for tests of significance.     

Functional properties of privalov spaces of holomorphic functions in several variables

We consider Privalov classes of degreeq>1 in the unit ball and the polydisk in ℂⁿ. They are defined, say, for the ball, as the sets of functionsf(z) such that the average of ln ₊ ^q |f(z)| over a sphere centered at the origin remains bounded as the radius increases to 1. These classes, which were introduced (in the one-dimensional case) by Privalov before 1941, were often used in the foreign literature in the last 10–20 years; typically, the notation varied and Privalov was not mentioned. We discuss various equivalent definitions of these classes as well as the most general properties, such as growth estimates, properties of the natural metric, and boundedness or total boundedness of subsets. Translated fromMatematicheskie Zametki Vol. 65, No. 2, pp. 280–288, February, 1999.     

The cone of quasi-semimetrics and exponent matrices of tiled orders

Finite quasi semimetrics on n can be thought of as nonnegative valuations on the edges of a complete directed graph on n vertices satisfying all possible triangle inequalities. They comprise a polyhedral cone whose symmetry groups were studied for small n by Deza, Dutour and Panteleeva. We show that the symmetry and combinatorial symmetry groups are as they conjectured.Integral quasi semimetrics have a special place in the theory of tiled orders, being known as exponent matrices, and can be viewed as monoids under componentwise maximum; we provide a novel derivation of the automorphism group of that monoid. Some of these results follow from more general consideration of polyhedral cones that are closed under componentwise maximum.     

Restrictions and extensions of potentials of Hp distributions

In this note we consider extensions and restrictions of anisotropic classes of distributions which include the usual global or homogeneous Lipschitz classes Λ(B, X) as well as the Sobolev classes $\text{L}{}_{\mathrm{\alpha }}{}^{\mathrm{p}}$. The novelty of our approach is that our results remain valid for all values of p (0 < p < ∞) as well as the critical index $\text{α =}\text{1}\text{p}$.     

Mathematical analysis of investment systems

Investment systems are studied using a framework that emphasize their profiles (the cumulative probability distribution on all the possible percentage gains of trades) and their log return functions (the expected average return per trade in logarithmic scale as a function of the investment size in terms of the percentage of the available capital). The efficiency index for an investment system, defined as the maximum of the log return function, is proposed as a measure to compare investment systems for their intrinsic merit. This efficiency index can be viewed as a generalization of Shannon's information rate for a communication channel. Applications are illustrated.     

Linear relations of refined enumerations of alternating sign matrices

In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices where in addition some left and right columns are fixed. The main result is a simple linear relation between the number of n×n alternating sign matrices where the top row as well as the left and the right column is fixed and the number of n×n alternating sign matrices where the two top rows and the bottom row are fixed. This may be seen as a first indication for the fact that the refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows as well as left and right columns can possibly be reduced to the refined enumerations where only some top and bottom rows are fixed. For the latter numbers we provide a system of linear equations that conjecturally determines them uniquely.     

Approximate Solutions of Continuous Dispersion Problems

The problem of positioning p points so as to maximize the minimum distance between them has been studied in both location theory (as the continuous p-dispersion problem) and the design of computer experiments (as the maximin distance design problem). This problem can be formulated as a nonlinear program, either exactly or approximately. We consider formulations of both types and demonstrate that, as p increases, it becomes dramatically more expensive to compute solutions of the exact formulation than to compute solutions of the approximate formulation.     

Asymptotic behavior of unbounded solutions of linear Volterra integral equations

The asymptotic behavior as t → ∞ of solutions of ∝₀^tu(t ? s) dA(s) = f(t) is studied when f(t) satisfies a “o” estimate as $\text{t ” ∞}$, and A belongs to a weighted space and its Laplace-Stieltjes transform has finitely many zeros in its closed half-plane of convergence. Results for systems of integral equations as well as for integrodifferential systems are also given.     

Projections of Bodies and Hereditary Properties of Hypergraphs

We prove that for every n-dimensional body K, there is a rectangularparallelepiped B of the same volume as K, such that the projectionof B onto any coordinate subspace is at most as large as thatof the corresponding projection of K. We apply this theorem to projections of finite set systems andto hereditary properties. In particular, we show that everyhereditary property of uniform hypergraphs has a limiting density.     

Analytical and numerical aspects of a generalization of the complementary error function

In this paper we discuss analytical and numerical properties of the function , with α,β,Rz>0, which can be viewed as a generalization of the complementary error function, and in fact also as a generalization of the Kummer U-function. The function V_ν,μ(α, β, z) is used for certain values of the parameters as an approximate in a singular perturbation problem. We consider the relation with other special functions and give asymptotic expansions as well as recurrence relations. Several methods for its numerical evaluation and examples are given.     

Examples of Calabi–Yau Threefolds as Covers of Almost-Fano Threefolds

We give a method for producing examples of Calabi–Yau threefolds as covers of degree d ≤ 8 of almost-Fano threefolds, computing explicitely their Euler– Poincaré characteristic. Such a method generalizes the well-knownclassical construction of Calabi–Yau threefolds as double covers of the projective space branched along octic surfaces.     

A classification of 4-connected graphs

Having observed Tutte's classification of 3-connected graphs as those attainable from wheels by line addition and point splitting and Hedetniemi's classification of 2-connected graphs as those obtainable from K₂ by line addition, subdivision and point addition, one hopes to find operations which classify n-connected graphs as those obtainable from, for example, K_n+1. In this paper I give several generalizations of the above operations and use Halin's theorem to obtain two variations of Tutte's theorem as well as a classification of 4-connected graphs.     

Rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping

We study the rate of decay of solutions of the wave equation with localized nonlinear damping without any growth restriction and without any assumption on the dynamics. Providing regular initial data, the asymptotic decay rates of the energy functional are obtained by solving nonlinear ODE. Moreover, we give explicit uniform decay rates of the energy. More precisely, we find that the energy decays uniformly at last, as fast as 1/(ln(t+2))^2−δ,∀δ>0, when the damping has a polynomial growth or sublinear, and for an exponential damping at the origin the energy decays at last, as fast as 1/(ln(ln(t+e²)))^2−δ,∀δ>0.     

Lower bound for the convergence rate of nonstationary Jacobi-like iteration

Stationary and nonstationary Jacobi-like iterative processes for solving systems of linear algebraic equations are examined. For a system whose coefficient matrix A is an H-matrix, it is shown that the convergence rate of any Jacobi-like process is at least as high as that of the point Jacobi method as applied to a system with 〈A〉 as the coefficient matrix, where 〈A〉 is a comparison matrix of A.